Frames, Erasures, and Signal Estimation with Stochastic Models
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Frames, Erasures, and Signal Estimation with Stochastic Models Somantika Datta1
Received: 22 November 2018 / Accepted: 17 December 2019 © Springer Nature B.V. 2019
Abstract Frame properties and conditions are determined that would minimize the error in signal reconstruction or estimation in the presence of noise and erasures. The special focus here is on stochastic models. These include estimating a random signal with zero mean and a general covariance matrix, minimizing the mean-squared error (MSE) when the frame coefficients are erased according to some a priori probability distribution in the presence of random noise, and also studying the use of stochastic frames in estimating a random signal. In estimating a random signal from noisy coefficients, when a frame coefficient is lost or erased, it is established that the MSE is minimized under certain geometric relationships between the frame vectors and the signal. When the coefficients are erased according to some a priori distribution, conditions are found for the norms of the frame vectors in terms of the probability distribution of the erasure so that the MSE is minimized. Results obtained here also show how using stochastic frames can lead to more flexibility in design and greater control on the MSE. Keywords Erasures · Estimation · Frames · LMMSE estimation · Random signal · Stochastic frames Mathematics Subject Classification (2010) 94A12 · 42C15
1 Introduction and Background Frame theory has gained significant attention in recent years due to applications in signal processing. Representation of signals in terms of frames are robust to transmission losses and resilient to noise. The purpose of this work is to determine conditions on frames so that signal reconstruction can be efficient in the presence of erasures. The study of optimal frames for erasures has been done by other authors, most notably in [5] and [4] where the latter also considers the presence of random noise in the frame measurements. Under this
B S. Datta
[email protected]
1
Department of Mathematics, University of Idaho, Moscow, ID 83844-1103, USA
S. Datta
class of work, much of the focus in the literature has been on deterministic models only. In contrast, the work here is focused on stochastic models. These include considering a random signal, the frame coefficients being erased according to some a priori probability distribution in the presence of random noise, and also the use of stochastic frames. The analysis is almost entirely for the situation where there is a single erasure, i.e., one of the frame coefficients is lost. Where applicable, comments are made on the situation with more than one erasure. Under a stochastic model, the general idea is to minimize the mean-square error when the signal is reconstructed from frame measurements that are corrupted by noise and erasures. Some of the calculations involving random signals are inspired by the work in [7] where a random vector is reconstructed from its noisy projections onto low-dimensional subspaces and under subspace era
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